Monetary and Fiscal Policy Interaction with Various Degrees of Commitment*

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1 JEL Classification: classification: E63, C73 Keywods: monetay vs. fiscal policy inteaction, Game of Chicken, commitment, dynamic leadeship, asynchonous games, explicit inflation tageting onetay and iscal Policy Inteaction with Vaious Degees of Commitment* Andew HUGHES HALLETT Geoge ason Univesity and Univesity of St. Andews Jan LIBICH VŠB-Technical Univesity Ostava, La Tobe Univesity and CAA coesponding autho Pet STEHLÍK Univesity of West Bohemia, Plzeň Abstact Well befoe the global financial cisis, the long-tem tend in fiscal policy had aised concens about isks fo the outcomes of monetay policy. Ae feas of an unpleasant monetaist aithmetic justified? To povide some insights, this pape examines stategic fiscal-monetay inteactions in a novel game-theoy famewok with asynchonous timing of moves. It genealizes the standad commitment concept of Stackelbeg leadeship by making it dynamic. By letting playes move with a cetain fixed fequency, this famewok allows policies to be committed o igid fo diffeent peiods of time. We find that the infeio non-ricadian (active fiscal, passive monetay) egime can occu in equilibium, and that this is moe likely in a monetay union due to fee-iding. The bad news is that, unlike unde the static commitment of Sagent and Wallace (1981), this may happen even if monetay policy acts as leade fo longe peiods of time than fiscal policy. The good news is that unde some cicumstances an appopiate institutional design of monetay policy may not only help the cental bank esist fiscal pessue and avoid the unpleasant monetaist aithmetic, but also discipline excessively spending govenments. By acting as a cedible theat of a costly policy tug-of-wa, long-tem monetay commitment (e.g. a legislated inflation taget) may induce a eduction in the aveage size of the budget deficit and debt, and move the economy to a Ricadian (passive fiscal, active monetay) egime. oe boadly, this pape demonstates that ou game-theoetic famewok with dynamic leadeship can help to uniquely select a Paeto-efficient outcome in situations with multiple equilibia whee standad appoaches do not povide any guidance. 1. Intoduction The ecent financial cisis has shown that the bodeline between fiscal () and monetay () policies is fine than most academics and policymakes had thought. This has highlighted the impotance of undestanding the inteaction between the two policies. The idea that and policies might inteact goes back to iedman (1948), Tinbegen (1954), undell (1962) and Coope (1969). But until ecently most models used fo policy design teated each policy in isolation. The subsequent liteatue has * We ae gateful to Don Bash, fome goveno of the Reseve Bank of New Zealand, fo his valuable views and fo contibuting the case study in Section 7.3. We would also like to thank the audience of the 2012 Ameican Economic Association confeence and of seveal othe confeences and seminas fo thei useful comments. We gatefully acknowledge financial suppot fom the Czech Science oundation (GA402/10/1046, GACR S), the Euopean Social und (CZ.1.07/2.3.00/ ) and the NKJ (Neuon) oundation. We also appeciate the detailed suggestions by the edito and thee anonymous efeees. 2 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

2 concentated on the govenment s diect institutional inteventions into cental banking. 1 The focus of this pape is indiect inteaction, which is moe subtle and less well undestood. It woks though spilloves of economic outcomes vaiables such as inflation, output and debt ae affected by both policies, and thei inteactions in tun affect the optimal setting of both policies. 2 We examine an inceasingly impotant aspect of the indiect policy inteaction that standad maco models do not captue, namely stategic policy behavio. Thee exists an abundance of ecent eal-wold examples of stategic inteactions between and policies. To name just one, despite the Euopean Cental Bank s initial distaste fo any fom of quantitative easing in the aftemath of the 2008 cisis, the ECB subsequently engaged in such actions, at least patly due to political pessues. To fill the gap in the liteatue we will analyze the stategic inteactions and possible policy spilloves though non-coopeative game-theoetic techniques. Let us stess fom the outset that ou inteest lies in medium to long-tem outcomes of the policy inteactions. We will not examine the optimal shot-tem mix of policy esponses to a shock such as the global financial cisis, and hence do not offe any assessment of the cuent stimulus vs. austeity debate. Ou analysis descibes the undelying long-tem stance of and policies, and ou policy implications theefoe only apply once economies have ecoveed fom the cuent global downtun. Ou pespective thus follows Sagent and Wallace (1981), Alesina and Tabellini (1987), Leepe (1991), Nodhaus (1994) and the subsequent liteatue, and links to cuent debates about sustainability, e.g. Kotlikoff and Buns (2012) and Leepe (2010). We fist motivate the analysis by outlining the lage size of the gap (unfunded liabilities) facing most advanced counties, pimaily due to aging populations and ballooning health-cae costs. We then examine unde what cicumstances these excesses may spill ove to policy. In this long-tem sense, we endogenize Leepe s (1991) policy egimes by deiving the cicumstances unde which we ae likely to obseve the Ricadian egime (active, passive ), and those unde which the unpleasant monetaist aithmetic (passive, active ) is likely to occu. Ou analysis contibutes to both macoeconomic policy and game theoy. On the game-theoy font, we use a novel famewok with genealized timing developed in Libich and Stehlík (2010), and employed also in Libich and Stehlík (2011) and Libich (2011). The policymakes no longe move simultaneously evey peiod. i, can change its medium-un stance with a constant Instead, each policy i fequency evey peiods (afte a synchonized initial move). igue 1 in Section 3 offes an illustation of such timing. 1 This includes the ability of the govenment ( policymake) to affect policy outcomes though the appointment of the cental banke (Rogoff, 1985), an optimal contact with the cental banke (Walsh, 1995), o though oveiding the cental banke (Lohmann, 1992). 2 Examples of this type of indiect inteaction include Leepe (2010), Cochane (2009), Adam and Billi (2008), Eusepi and Peston (2008), Chadha and Nolan (2007), Pesson et al. (2006), Benhabib and Eusepi (2005), Gali and onacelli (2005), Hughes Hallett and Weymak (2007, 2005), Eggetsson and Woodfod (2006), Dixit and Lambetini (2003, 2001), van Aale, et al. (2002), Leepe (1991) and Sagent and Wallace (1981). inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 3

3 We intepet the numbe of peiods as the degee of igidity (which aguably depends on the size of the gap). In contast, we intepet as the degee of long-tem commitment, as it descibes the cental bank s inability to change its taget fo aveage inflation. 3 This famewok is a genealization of the altenating move games of askin and Tiole (1988) and Lagunoff and atsui (1997) fo which the existing wok povides a stong motivation. o example, Cho and atsui (2005) ague that: [a]lthough the altenating move games captue the essence of asynchonous decision-making, we need to investigate a moe geneal fom of such pocesses [...]. Allowing fo leads to asynchonous policy moves with each playe acting as a Stackelbeg leade fo at least some pat of the stage game. 4 We show that conventional conclusions made unde the standad commitment concept of Stackelbeg leadeship, which is static, may not be obust. This is because it cannot captue the cost of the policy conflict o mis-coodination the Stackelbeg followe moves immediately afte obseving the leades move, so thee ae no such costs. This highlights the impotance of incopoating the time dimension into the sequencing of policy actions: that is, the impotance of modeling the commitment dynamics and leadeship changes explicitly. On the policy font, the pape s contibution is to show that concens about long-tem excesses spilling ove to monetay policy may be justified. 5 We ae able to identify seveal vaiables and cicumstances which can make that happen. We then show, using the asynchonous stuctue of the game, how institutional emedies can pevent such spilloves unde some (but not all) cicumstances. Inteestingly, ou analysis implies that policy commitment can sometimes indiectly discipline policy, and achieve the socially optimal outcomes fo both policies thoughout the medium and long un. Intuitively, if the inflation taget is explicitly stated in the statutes o elated legislation, the cental banke is moe willing to engage in a costly tug-of-wa with the govenment, counteacting excessive actions with stong tightening. As this would eliminate any political gains, the govenment s incentive to engage in excessive actions o avoid tough efoms fades away, leading to an impovement in the budget and debt. 6 3 This inability is likely to depend on how explicitly the long-tem taget is gounded in the cental banking statues o legislation. It should be stessed that due to ou medium-tem focus should not be intepeted as the fequency of the cental bank s inteest-ate decisions. This is because does not estict the ability to make peiod-by-peiod policy stabilization decisions egading zeo mean shocks. It only esticts the fequency with which the aveage inflation level can be alteed. 4 This captues the obsevation of Tobin (1982) that some decisions by economic agents ae econsideed daily o houly, while othes ae eviewed at intevals of a yea o longe [...], and follows Tobin s call: [...] it would be desiable in pinciple to allow fo diffeences among vaiables in fequencies of change [...]. 5 Let us state again that we ae inteested in stuctual imbalances, not cyclical ones caused by tempoay actions in esponse to cyclical downtuns. 6 In elation to that, Section 7.3 pesents a shot case study witten by D. Don Bash, goveno of the Reseve Bank of New Zealand fom 1988 to His contibution descibes the developments in New Zealand shotly afte the adoption of an explicit commitment to a low-inflation taget and highlights the disciplining effect this aangement has had on policymakes. This effect seems obust as it was deived in Libich and Stehlík (2011) in a diffeent timing famewok featuing stochastic athe than deteministic moves. Empiical evidence on this effect using time-vaying paametes VARs is pesented in anta, Libich, and Stehlík (2012) and discussed below. 4 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

4 Thee is one impotant caveat to this conclusion, namely membeship in a cuency union. We show fomally in Section 6 how accession to a cuency union may intoduce a fee-iding poblem. Intuitively, if a small membe county engages in pofligacy, its impact on the inflation outcomes of the union as a whole is small. Because of that, the punishment by the common cental bank will be of a much smalle magnitude. uthe, it will be spead acoss all membe counties (see asson and Patillo, 2002). Theefoe, if a small, fiscally iesponsible county does not intenalize the cost it imposes on othes, it tends to spend excessively moe so than befoe joining the union, when it would have bone the full weight of its own cental bank s punishment. Developments in the Euozone in the lead-up to the global financial cisis povide an example of that sot of phenomenon (although this is clealy not the only souce of the cuent poblems). This constitutes a diffeent type of moal hazad to the one commonly discussed with espect to union accession, whee one county elies on a bailout by the est of the membes. 2. Stategic Policy Inteactions in the Pesence of a iscal Gap This section lays out the simplest maco setup that can captue the main featues of the long-tem stategic policy inteaction in the pesence of integeneational imbalances. An advantage of such an appoach is the fact that ou game-theoetic insights ae not model specific and can be applied acoss a wide ange of diffeent monetay-fiscal inteaction models in the liteatue. A disadvantage of ou appoach is an ovesimplification of the macoeconomic envionment in seveal espects, which we believe is justified due to ou focus on the game-theoetic analysis. 2.1 Budget Constaint Ou exposition of inte-tempoal issues builds on Leepe and Walke (2011), whose model featues a flow budget constaint in the fom B I 1B 1 Z T (1) L t t t t t t t Lt whee B is the stock of govenment bonds (debt) in nominal tems that pay a goss nominal etun of I, and L is the pice level. Z t T t is the (eal) level of net tansfes (tansfes Z minus taxes T) pomised to households by the existing legislation. Howeve, the actual (deliveed) level of futue net tansfes is λ t (Z t T t ). This 1 0,1 is a eneging paamete, and that λ t (Z t T t )L t expesses implies that t the deliveed net tansfes in nominal tems ( dollas ). Intuitively, (1) states that existing debt including inteest payments must be paid fo by futue pimay supluses o, up to a point, by issuing new bonds. uthemoe, pomised net tansfes can be eneged upon by the govenment (passive policy), o thei eal value inflated away by the cental bank (passive policy). The latte solution has been intepeted in a numbe of ways, most commonly as the unpleasant monetaist aithmetic of Sagent and Wallace (1981) o Leepe s (1991) fiscal theoy of the pice level. Due to ou focus on the game-theoetic analysis, we will suppess the dynamics (which have only second-ode effects). Specifically, we will in this section conside inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 5

5 Tabel 1 Net Pesent Value of the Impact on the iscal Deficit of the 2008 Cisis and Aging-Related Spending Respectively, as % of GDP Souce: I (2009). County Cisis Aging Austia Canada ance Gemany Italy Japan Koea exico Spain Tukey United Kingdom United States Advanced G-20 counties two peiods only, whee t 1 in (1) will epesent the past and t will epesent the futue. The fact that in most advanced counties long-tem stance is unsustainable seems uncontovesial; fo some discussion see, fo example, Kotlikoff and Buns (2012) o Leepe (2010). To povide just one piece of data, Table 1 epots I (2009) estimates of the net pesent value of the impact of aging-elated spending and of the 2008 financial cisis (as a pecentage of GDP). While the latte ae non-negligible, they ae dwafed by the fome leading to a lage pedicted gap. 7 The simplest way to incopoate such a gap is to impose: Z t T > 0 (2) t 2.2 Policy Objectives Befoe we fomally define the active and passive policy stance, we need to postulate the policymakes utility functions. This is done in a way consistent with the standad intuition of the dynamic policy ules of Leepe (1991): whee i, 2 2 T Bt T 2 t i t i1 t Lt U L L b denotes the policymakes, L T is the optimal inflation taget, b T the taget fo eal debt, i 0 epesents the degee of thei inflation consevatism and 0 is thei avesion to eneging upon pomised net tansfes, both elative to i 7 o example fo the United States, Batini et al (2011) povide a ecent estimate of the fiscal gap and conclude that: [...] unde ou baseline scenaio, a full elimination of the fiscal and geneational imbalances would equie all taxes to go up and all tansfes to be cut immediately and pemanently by 35 pecent. A delay in the adjustment makes it moe costly. (3) 6 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

6 debt vaiability (the middle tem of the utility function). 8 This implies that while the policy tagets ae the same (fo pasimony), the policy weights on these tagets may diffe between playes. Let us define a esponsible policymake as one with i i. In this pape we will estict ou attention to the elevant case of a esponsible cental bank (4) 2.3 Active/Passive Policies Next we adapt Leepe s (1991) teminology to ou long-un envionment. Definition 1: An active policy stance, A, povides no adjustment to balance the budget constaint (1). In contast, a passive policy stance, P, is a level ( * * Lt fo and t fo ) that povides the full adjustment necessay to balance the budget constaint and keep stable eal debt independently of the othe policy (i.e. assuming the othe policy is active). Using the A and P dichotomy we can analyze the stategic aspect of the policy inteaction as a 2 2 game. 2.4 Game-Theoetic Repesentation The payoff matix below summaizes the geneal game with {a, b, c, d, v, w, y, z} denoting the policymakes payoffs in the fou possible policy egimes. 9 Passive Active P A Active Ricadian Explosive (5) A a, v b, w Passive is-coodination Non-Ricadian (unpleasant aithmetic) P c, y d, z The payoffs ae obviously functions of the deep paametes of the undelying macoeconomic stuctue and policy pefeences, in ou case equations (1) (4). 10 In tuncating the setup to a 2 2 game we will follow Backus and Diffill (1985) and Cho and atsui (2005) and offe a tanspaent example in which the numbe of fee paametes is educed by nomalizing: T T t1 t1 1 1 B I L b T 1Z Note that debt vaiability is closely positively elated to output vaiability, which is a standad component of educed-fom policy pefeences. 9 As usual, the fist payoff efes to the ow playe () and the second to the column playe (). 10 uthe examples of how the linkages between maco models and the game-theoetic epesentation wok can be found in the woking pape vesion of this aticle and in a diffeent context in Libich and Stehlík (2010). inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 7

7 We focus on the futue dives of fiscal stess athe than past dives by stating off with initial debt on taget. 11 To deive the P policy level fom Definition 1, we use this nomalization in the matix in (5) togethe with L t = L T, which yields * * Bt Bt 1 and implies t 0. Similaly, the P policy level Lt is obtained fom (1) Bt T by imposing t 1 and b 1 L. This implies * Lt 2. Combining these with t (1) (2) yields the following debt outcomes in the fou policy egimes: A P * P t 0 A t 1 Stable eal debt Rising eal debt Bt 1 T b L 1 Bt 1.5 T b L 1 (6) t Stable nominal debt Rising nominal debt alling eal debt Stable eal debt Bt 1 T b L 2 Bt 2 L 2 b t Stable nominal debt Rising nominal debt t t T In the Ricadian and non-ricadian egimes [to use Woodfod s (1994) teminology] the gap is dealt with by the and policies espectively, and the eal debt buden is stable. By contast, in the (A, A) egime neithe policy deals with the poblem, and hence eal debt is on an explosive path. inally, in the (P, P) egime both policies deal with the poblem without coodination and hence eal debt falls excessively. 2.5 Possible Policy Scenaios Using (6) with (1) (3) and Definition 1 we can deive the policymakes payoffs: P A, P 1 1,, 4 4 A 1 1, (7) 4 4 Seveal scenaios, epoted in Table 2, ae possible depending on the policy weights. 11 Nevetheless, all ou esults apply to the case in which the existing debt is excessive and needs to be B educed, i.e. 0 T. L b 0 8 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

8 Table 2 Desciption of Possible Policy Scenaios Scenaio Pue Nash ixed Nash Coodination Poblem Policy Conflict Symbiosis (A, P) no no no Discipline (A, P) no no no Paamete Range min 4 Tug-of-wa (A, A) no no yes, Pue Coodination Game of Chicken (A, P) (P, A) (A, P) (P, A) yes yes no yes yes yes Neglect (P, A) no no no 1 max, 4 max, The scenaios diffe in the pobability of spilloves onto policy (in ascending ode). 1. The Symbiosis scenaio (using Dixit and Lambetini s teminology) wheeby the (A, P) outcome is the unique Nash equilibium and also both playes most pefeed outcome. Thus spilloves do not occu. 2. The Discipline scenaio in which the (A, P) outcome is still the unique Nash equilibium. Hence the spilloves ae avoided, but the outcome is no longe s most pefeed outcome. 3. The Tug-of-Wa scenaio wheeby the (A, A) outcome is the unique Nash equilibium. While spilloves will not occu in the medium un, this egime cannot obtain in the long un as the govenment s budget constaint (1) is not satisfied The Pue Coodination scenaio featuing two pue Nash equilibia, (A, P) and (P, A), and a mixed stategy Nash equilibium that is Paeto infeio to both pue Nash equilibia. spilloves onto policy may o may not occu, but as both policymakes pefe the fome pue Nash equilibium they ae unlikely. 5. The Game of Chicken scenaio similaly featues two pue Nash equilibia, (A, P) and (P, A) and one in mixed stategies. But in this case spilloves onto policy ae moe likely as each policymake pefes a diffeent pue Nash equilibium. 6. The Neglect scenaio in which (P, A) is the unique Nash equilibium, so spilloves suely occu. 12 In an impotant body of wok, Davig and Leepe (2011) examine the combination of (A, P) such as in Agentina in 2001 and (P, A) as many fea holds in the Geat Recession situation that eplaces the (A, A) outcome when the economy appoaches/hits its limit. inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 9

9 o thee easons, ou attention will be pimaily diected at the Game of Chicken summaized by 13 : 1 1, and, 4 4 (8) ist, it is the most inteesting scenaio fom the game-theoetic point of view as thee ae equilibium selection poblems. Specifically, neithe standad no evolutionay game-theoetic techniques can povide a clea choice between the pue Nash equilibia due to the symmety of the game. It is also the only scenaio of the six in which the timing of the actions detemines the equilibium. Unde the standad commitment concept the Stackelbeg leade will ensue its pefeed outcome, wheeas in the othe five scenaios leadeship does not alte the equilibium play. Second, the game featues both a conflict (each playe ties to secue its pefeed pue Nash equilibium) and a coodination poblem (to avoid the infeio mixed Nash). These two chaacteistics seem to occu in many eal-wold cases as well as in a wide ange of policy inteaction models: see Leepe (2010), Adam and Billi (2008), Banch et al. (2008), Resende and Rebei (2008), Benhabib and Eusepi (2005), Dixit and Lambetini (2003, 2001), Banett (2001), Bhattachaya and Haslag (1999), Atis and Winkle (1998), Blake and Weale (1998), Nodhaus (1994), Sims (1994), Woodfod (1994), Leepe (1991), Wyplosz (1991), Petit (1989), Alesina and Tabellini (1987), and Sagent and Wallace (1981). The intuition of ou findings will theefoe apply to any of these divese settings. 14 Thid, in this scenaio neithe policy s pefeences ae stongly skewed towads one objective. In paticula, even a situation of a consevative cental bank (with ) and a govenment putting moe weight onto debt stabilization than buying votes (with ρ close to 0) will fall into this categoy. Put diffeently, in the non-nash (A, A) and (P, P) egimes eal debt is on an unstable path and thus these cannot be equilibia fom a long-tem pespective. 3. Dynamic Commitment This section pesents an asynchonous game famewok with genealized timing of actions as developed in Libich and Stehlík (2010). Ou goal is to examine how the medium-un macoeconomic outcomes of the policy inteaction depend on this timing and othe vaiables. 3.1 Timing Afte a synchonized initial move in peiod t 1 each playe i moves with i a cetain constant fequency, namely evey peiod. igue 1 offes an example of such timing. 13 The emaining scenaios of Table 2, and altenative ones in which (4) does not hold, ae discussed in Section Switching between the two egimes is a possibility in many of these papes and analogous to ou mixed Nash equilibium. Theefoe, while the pape does not offe a welfae citeion to ank the two pue Nash equilibia, it is desiable to deive cicumstances fo unique equilibium selection ensuing that the Paetoinfeio mixed Nash is avoided. 10 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

10 igue 1 An Asynchonous Stage Game with Deteministic Commitment/Rigidity: Illustation of the Recusive Scheme and the Concepts of i, R, k and n i unde = 3, = 5 The vaiable i can be intepeted as the degee of commitment o igidity of playe i. These two concepts ae fomally identical in ou famewok, both efeing to the playes inability to move. Nevetheless, in the eal wold such inability comes fom diffeent souces, which we will acknowledge by efeing to as long-tem commitment and to as igidity. The degee of the latte is aguably closely linked to the size of the county s gap, Z t T t, since many govenment outlays elated to aging populations ae mandatoy. In tems of the fome, let us stess again that does not denote the fequency of the cental bank s inteest ate decisions in ou medium-un envionment. It should be intepeted as the fequency with which the cental bank can alte its taget fo aveage inflation (selected based on stategic consideations and may thus diffe fom L T ). This fequency typically depends on how explicitly its commitment to the taget is legislated. Because the taget only has to be achieved on aveage ove the business cycle, it does not have to be stict in the Rogoff (1985) sense; moe discussion of this follows in Section Assumptions and Notation o maximum compatibility ou famewok adopts all the assumptions of a standad epeated game. ist, commitment and igidity i ae exogenous and constant thoughout each game. Second, they ae common knowledge. Thid, all past peiods moves can be obseved. outh, the game stats with a simultaneous move. ifth, the playes ae ational, have common knowledge of ationality and, fo expositional claity, they have complete infomation about the stuctue of the game and the opponent s payoffs. These assumptions can easily be elaxed (some altenatives, such as endogenous i, will be discussed below). They ae intoduced hee so that the only diffeence fom the standad epeated game is in allowing i > 1 values that diffe acoss playes. This specification implies that the stage game of ou asynchonous setup is itself an extensive-fom game lasting T peiods, whee T is the least common multiple of and. o instance, the dynamic stage game in igue 1 with = 5 and = 3 is T = 15 peiods long. Denoting n i to be the i s playe s n th move (not peiod), and N i the numbe of T, i moves in the asynchonous stage game, it follows that N i. Also, l n inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 11

11 and l n will denote a cetain action l A, P A in the govenment s second move. o the est of this section we assume some at a cetain node n i ; e.g. 2 i j A efes to without loss of geneality, whee i, j. We can then denote 1 to be the playes elative j commitment/igidity. Also, i will be the intege value of elative commitment (the floo) j and i i R 0,1 (9) j j denotes the factional value of elative commitment (the emainde). 15 It will be evident that R plays an impotant ole as it detemines the exact type of asynchony in the game. Note also that if R 0 both playes take leadeship at some point duing the stage game. P A uthe, we denote B(.) to be the best esponse. o example, 1 B 1 P A is s best esponse to s initial A move, and 1 B 1 P expesses that 1 expesses that it is the unique best esponse. inally, an asteisk will denote an optimal play, e.g. 1 B1 expesses that s optimal play in move 1 is the best * esponse to s fist move. 3.3 Recusive Scheme The fact that we will be able to pesent poofs fo geneal values of i s due to the existence of a ecusive scheme in the moves. omally, let k n be the numbe of peiods between the n i -th move of playe i and the immediately following move of playe j (fo a gaphical demonstation see igue 1). Using this notation we can summaize the ecusive scheme of the game as follows: k j j kn R if kn R n1 j j kn 1 R if kn R Geneally, kn is a non-monotonic sequence, and we always have k Histoy, utue, Stategies and Equilibia By convention, histoy in peiod t, h t, is the sequence of actions selected pio to peiod t. And the futue in peiod t is the sequence of cuent and futue actions. It follows fom ou pefect monitoing assumption that ht is common knowledge at t. Let us efe, following Aumann and Soin (1989), to moves in which a cetain action l A, P is selected fo all possible histoies as histoy-independent. i j (10) 15 o example, in igue 1we have 5 2 1, and. 3 R 3 12 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

12 i A stategy of playe i is a vecto that, ht, specifies the playe s play n. The asynchonous game will commonly have multiple Nash equilibia. To select among these we will use a standad equilibium efinement, subgame pefection, that eliminates non-cedible theats. Subgame pefect Nash equilibium (SPNE) is a stategy vecto (one stategy fo each playe) that foms a Nash equilibium afte any histoy ht. Given the lage numbe of nodes in the game, we focus on the equilibium path of the stage game SPNE, i.e. the actions that would actually get played. In doing so, we will use the following teminology: Definition 2: (i) Any SPNE of the asynchonous stage game that has, thoughout its whole equilibium path, the Ricadian egime (A, P) will be called a Ricadian SPNE. The case in which all SPNEs of the game ae Ricadian will be called a Ricadian Wold. (ii) Any SPNE with the non-ricadian egime (P, A) thoughout its whole equilibium path will be called a non-ricadian SPNE. The case in which all SPNEs of the game ae Ricadian will be called a non-ricadian Wold. (iii) Any SPNE othe than Ricadian and non-ricadian will be called a Regimeswitching SPNE. The case in which thee exist multiple types of SPNE (any combination of Ricadian, non-ricadian and Regime-switching) will be called a Regime-switching Wold. 3.5 (Non)-Repetition As ou focus is on conditions unde which an efficient outcome uniquely obtains on the equilibium path of the asynchonous stage game, its futhe epetition can be disegaded without loss of geneality. Intuitively, if the effective minimax values of the playes in the dynamic stage game [that ae the infima of the playes subgame pefect equilibium payoffs; see Wen (1994)] ae unique and Paetoefficient, then the effective minimax values of the epeated game (with any finite o infinite numbe of epetitions) will also be the same. Put diffeently, the set of Paetosupeio payoffs is empty as we ae aleady on the Paeto fontie. The uniqueness popety also implies that we can focus on pue stategies only, without loss of geneality Results in the Game of Chicken without Discounting It is impotant to ecall Section 2, in which we showed that the Game of Chicken type inteactions aise unde some (but not all) paamete values depending on the model and policy pefeences. To develop the intuition of the game-theoetic famewok, this section epots esults fo the specific payoffs in (7) (8) deived fom the maco setup and fully patient playes with discount factos δ = δ = 1. Section 5 will then allow fo the policymakes impatience and solve the game fo the geneal Game of Chicken payoffs in (5), namely, and, ad max bc z v max wy (11) 16 Repetition is commonly used to help alleviate inefficiency and enhance coopeation though eputational channels (see e.g. ailath and Samuelson, 2006), o in the monetay context (Bao-Godon, 1983). The advantage of focusing on the dynamic stage game is that it povides the wost-case scenaio in which eputation cannot help secue coopeation. inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 13

13 igue 2 The Space eatuing the Thesholds and Regions of SPNE Poposition 1: Conside the Game of Chicken (7)-(8) without discounting. (i) The Ricadian Wold occus if and only if commitment is sufficiently stong elative to igidity,,, (12) (ii) The non-ricadian Wold occus if and only if commitment is sufficiently weak elative to igidity,,, (13) + + (iii) The Regime-switching Wold occus if and only if commitment is neithe sufficiently stong no sufficiently weak elative to igidity, (14) Poof: Appendix A (on the website of this jounal) deives the exact fom of the necessay and sufficient thesholds and. igue 2 summaizes these esults gaphically. The space is expanded compaed to a one-shot (o simultaneously epeated) game whee 1, and the static concept of commitment (Stackelbeg leadeship) whee 0,. The space can now be boken into thee main equilibium egions as defined in Definition 2. Note that, in contast to Stackelbeg leadeship, ou famewok gives us additional valuable infomation. Among othe things, it tells us the exact degee of commitment/igidity equied if a playe s pefeed outcomes is to be achieved. The theshold is a function of seveal vaiables: see (16) below fo an example. uthemoe, it shows that once iche dynamics/leadeship ae allowed thee may still be egime switches. This is unlike the static Stackelbeg leadeship whee the intemediate egion does not exist. Let us sketch the intuition behind this faily complex poof using the simple case 2, 1 (which is a special case of R = 0). o the cental bank to ensue 14 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

14 the Ricadian Wold obtains, it must be willing to engage in a costly tug-of-wa. That is, the bank s incentive compatibility elation between the costs and benefits of alte- A A native couses of action, B1 1 b a d, has to hold (15) (A, A) (A, P) (P, A) Policy victoy suendeing conflict Reaanging, and using the payoffs in (7) (8), yields the following theshold value conflict cost R R (16) victoy gain If this condition is satisfied then the cental bank s victoy ewad moe than offsets its initial conflict cost, and hence the bank is not willing to accommodate excessive policy. Such detemination to fight if necessay eliminates the incentive of to un stuctual deficits and accumulate debt. In othe wods, since the theat of a tug-of-wa is cedible, thee is in fact no such policy conflict in equilibium as suendes fom its initial move. The implication is that in a Game of Chicken sufficiently stong commitment has two consequences. Not only is it capable of shielding the cental bank fom pessue and spilloves, but it can also discipline policy by impoving the govenment s incentives and equilibium play. Section 7.3 pesents a shot case study by D. Don Bash documenting how this actually happened in New Zealand. He explains that adoption of a stonge commitment gave him, as goveno, moe ammunition to stand up to excessive policy; and that this in tun had a disciplining effect on policymakes. then the victoy ewad is insufficient to compensate fo the initial conflict cost with.the theat of not accommodating policies is no longe fully cedible, and thus we move fom the Ricadian Wold to the Regime-switching Wold in which spilloves may occu. If the degee of In contast, if, commitment (elative to the size of the gap) is even lowe, then we ente the non-ricadian Wold in which spilloves suely occu. It is now who is willing to engage in a costly tug-of-wa with. The poof of Poposition 1 (Appendix A) also shows that the natue of this special case R = 0 caies ove to moe geneal asynchonous cases with R > 0. This may appea supising because in the latte case both playes take the ole of inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 15

15 the leade duing the dynamic stage game. o example, in igue 1 thee ae fou changes in leadeship (in 2, 2, 3, and 3 ) and because of that thee ae multiple peiods of potential policy conflict with the decisions about them intetwined. The intuition fo this esult is twofold. ist, fo any R, the playe with lowe i makes the last evision which can be exploited by the opponent. Second, the most impotant action happens in the initial simultaneous move, since the conflict cost is at its maximum and would last the longest elative to the victoy ewad. This initial move theefoe yields the stongest incentive compatibility condition. omally poving this esult which is not obvious by any means is one of the methodological contibutions of this pape. This esult also highlights the usability of ou asynchonous game: uses can focus on the stat of the game and ignoe the est of the complicated dynamics. Put diffeently, uses can utilize the special R = 0 case. The findings ae in contast to those unde the standad Stackelbeg commitment, wheeby the leade (committed playe) wins the game (gets its pefeed outcome) independently of any stuctual o policy paametes. The esults of Poposition 1 can be viewed as a efined vesion of the conventional esult. They offe a possible explanation fo the obseved institutional diffeences acoss counties and a iche basis fo policy ecommendations. Coollay 1: (i) Geate avesion of the govenment to eneging on pomised net tansfes,, inceases the paamete egion of the non-ricadian Wold, and (togethe with the cental bank s eneging avesion ) deceases the egion epesenting a Ricadian Wold. (ii) A moe explicitly committed cental banke (highe ) can be less consevative (have lowe ) while still ensuing a Ricadian Wold. Claim (i) is intuitive: geate eneging avesion inceases the peceived (political) cost of efom and thus makes the non-ricadian solution to the gap poblem (and the unpleasant monetaist aithmetic) moe likely. Claim (ii) is pehaps supising as it implies patial substitutability of stict and explicit inflation tageting, but it can be seen in (12) whee is deceasing in. The moe explicitly committed the egime is, the less stict on inflation it has to be. Claim (ii) is at odds with concens by inflation tageting skeptics such as Geenspan (2003) and Kohn (2005), who believed that an explicit inflation taget educes the peiod by peiod policy flexibility needed to stabilize the eal economy. But ou finding is in line with Woodfod, who called such concens the taditional pejudice of cental bankes, and Svensson (2008), who agued that it is desiable to do flexible inflation tageting moe explicitly. Similaly, ecent empiical evidence of Pakin (2013), Kuttne and Posen (2011) and Ceel and Hubet (2010) shows that explicit inflation tagets have not led to sticte policy duing the global financial cisis. It is appaent that, in the Regime-switching Wold, the vaiability of (tend) inflation and debt is geneally highe than in a Ricadian Wold due to changes in the policy stance. Cycles in aveage inflation and debt can be geneated unde some cicumstances. We leave a moe detailed investigation of this intemediate egion fo 16 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

16 futue eseach and only epot one finding that qualifies the intuition of the standad Stackelbeg commitment in an impotant way. Poposition 2: Conside the Game of Chicken (7) (8) and some,. Despite the cental bank s acting as the leade fo longe peiods of time than the govenment, thee ae paamete values unde which the bank s pefeed Ricadian SPNE does not exist, but the govenment s pefeed non-ricadian SPNE does. Poof: Appendix B (on the web-site of this jounal) shows that this happens if and, i.e. when the cost of suendeing (passive policy) is sufficiently low fo the cental bank, but sufficiently high fo the govenment. The fact that the playe with a longe leadeship peiod (highe i ) cannot achieve its pefeed outcome, while the opponent can, is in stak contast with the standad Stackelbeg leadeship conclusion. It is useful, as a matte of methodology, to ecognize how this game with dynamic commitments has been solved. In a conventional dynamic game, an equilibium solution can be found by backwads ecusions to each decision point, taking account of any subsequent decisions to be made and conditional on the last known state of the system befoe that decision. But such an appoach is only valid if each playe s poblem can be witten as one of optimizing an additively ecusively sepaable constained objective function at each t, so that Bellman s (1961) pinciple of optimality applies. As an appoach, it beaks down if the constaints contain fowad looking behavio: fo example when ational expectations of futue outcomes ae pesent. Ou solution, fist applied in Libich and Stehlík (2010), also has that featue. This is because at each decision point, each playe has to know if a given stategy is cuently supeio to the othes and will continue to emain supeio long enough fo that playe to enjoy those gains long enough to dominate the payoffs available unde the othe stategies. The latte component of the calculation is the fowad looking element and will depend on the elative discount factos and the elative lengths of commitment/igidity (which detemine when the opponent has the next chance to west contol of the gains back to its own pefeed stategy). The methodology point hee is that fowad and backwad dependencies aise in dynamic models with ational expectations, of couse. Howeve, ou analysis shows that asynchonous games, even without explicit ational expectations in the constaints, ae a second case in point. 5. Extension I: Discounting and Geneal Payoffs This section intoduces discounting fo both playes, 1, 1, and solves the Game of Chicken fo geneal payoffs (11). It will become appaent that while the natue of the above game-theoetic analysis is obust to discounting, the playes impatience may change the outcomes in an impotant way. We will focus on deiving conditions of the Ricadian Wold. But as Poposition 1 demonstates, the esults apply analogously fo the non-ricadian one. inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 17

17 Poposition 3: Conside the geneal Game of Chicken (11) with discounting. The Ricadian Wold occus iff the policymake is both sufficiently patient d b a b and sufficiently stongly committed,,, a, bdvwyz,,,,, (18) If is insufficiently patient,, then commitment cannot ensue the Ricadian Wold. (17) does not exist, and even infinite Poof: See Appendix C (on the website of this jounal) that deives the exact fom of the necessay and sufficient theshold in (18), namely equation (39). 17 This implies that is a step function of s payoffs, specifically inceasing in v and y, and deceasing in w and z. In tems of the othe vaiables, while we cannot fomally pove the elationships fo all R, valuable insights can nevetheless be obtained fom the special case of R = 0, which was shown above to be epesentative of the moe asynchonous cases R 0. The theshold (0) is inceasing in, δ and d, and deceasing in δ, a and b: see igue 3 fo a gaphical demonstation. 18 Intuitively, s impatience stengthens the necessay and sufficient condition; it makes it moe difficult (and eventually impossible) fo to dominate and ensue a Ricadian Wold. The intuition is simila to a standad epeated game in which it is hade to dete an impatient playe fom defecting, since the futue ewad fo not defecting has asmalle pesent value. The policy implication is theefoe the following: a less patient cental banke needs to commit moe explicitly to guaantee its pefeed medium-tem outcomes. This implies patial substitutability between an explicit inflation taget and longe mandates fo cental banks. Poposition 3 does not only efine the standad esult obtained unde Stackelbeg leadeship, it also qualifies its intuition substantially. Unde static commitment, the (moe) committed playe always ensues its pefeed egime. Unde dynamic commitment thee ae paamete egions in which he may do so and paamete egions in which he neve does so. If the moe committed playe is highly impatient, then even an infinitely stong commitment cannot ensue its pefeed egime egadless of the opponent s discount facto. Hence the insights obtained unde the standad commitment concept ae not obust. 17 The patience and commitment thesholds fo the non-ricadian Wold ae again mio images of and. 18 omal poofs of these elationships appea in the woking pape vesion of this aticle. 18 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

18 igue 3 Dependence of () 0 on δ fo vaious fom the necessay and sufficient condition (31) fo the game in (7) (fo illustation using 2 and 0 ). The Ricadian Wold occus in the aea to the ight of the cuves. The dotted asymptotes coespond to the bounds () 0 fo each paticula. Relating this esult back to igue 2 if thee ae only non-ricadian and Regime-switching Wolds, and only the latte if and such case neithe no exist. 6. Extension II: iscal Heteogeneity in a onetay Union since in The debt cisis in the Euozone has ecently eceived a lot of attention. Ou dynamic commitment famewok can offe some insights as it can easily incopoate any numbe of playes. Let us examine the case in which policy is heteogeneous focusing on two types of heteogeneity: in economic size and in igidity. This descibes the situation in the Euopean onetay Union and, to some extent, the United States, with a common cuency and hence common policy, but somewhat independent policies. j omally, the set of playes is now I, a cetain membe county, that J s j j1 j its igidity, and s j whee j[1,, J] denotes its elative economic size such 1. We assume that the oveall payoff of is a weighted aveage of the bank s payoffs gained fom the inteaction with each j with weights sj. The payoff of each independent govenment is, howeve, diectly detemined by its own actions and those of the common cental bank as shown in (5). 19 In extending ou analysis to this case we will fist assume the absence of feeiding by union membes. In that case, the natue of the above esults emains unchanged. 19 Indiectly, the actions of othe govenments also have an impact since they detemine the action of the cental bank and hence the equilibium outcomes in each county. inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 19

19 Remak 1: In a monetay union without fee-iding, the necessay and sufficient theshold in the Game of Chicken is inceasing in the weighted aveage of the igidities of membe counties with the weights being the county sizes s j. omally, in (12) (13) and (17) (18) is eplaced by J s j j j1 Howeve, a moal-hazad poblem may occu on the pat of individual govenments. This is because the political benefits of spending accue pimaily to the fiscally iesponsible county, wheeas the economic costs in tems of tighte policy ae spead acoss all counties (asson and Patillo, 2002). In paticula, the smalle a county is elative to the union, the less impact its policy has on aveage inflation and output foecasts in the union and hence on the inteest ate esponse of the common cental bank. uthemoe, the punishment in the fom of an ensuing contaction is also spead acoss the union as a whole. Hence even disciplined govenments ae penalized. The incentives fo fee-iding, whethe delibeate o out of myopia o neglect, can theefoe ise apidly, especially fo the smalle counties. To fomalize this, denote by m j [0,1] the degee of fee-iding: i.e. the extent to which membe county j does not intenalize the negative impact of its excesses on the est of the union membes. The value mj = 0 denotes no fee-iding, wheeas mj = 1 denotes exteme fee-iding in which county j totally ignoes its impact on othes. The effect of this fee-iding can be incopoated in ou analysis though the payoff vj. That paamete epesents the govenment s suende payoff elative to the conflict cost. It seems natual to assume the j govenment s conflict cost to be inceasing in the county s weight s j, and deceasing in its degee of fee-iding m j :,, vj mj sj vj mj sj 0, and 0 fo all m j 0 m s j That is, if the govenment of membe county j decides to fee-ide, it will face (and intenalize) a smalle punishment fom the common cental bank than it would have encounteed fom the county s independent cental bank pio to joining the monetay union. Let us futhe assume that v j is a monotone function of m j, and that v m 1 w. We can now contast the outcomes in some county j befoe ( ) j j j and afte ( ) m j j j. 20 (19) joining the union. The case befoe joining the union is natually 0 and s 1. Afte joining, we have some m [0,1] and s (0,1). j j 20 To offe a numeical example, assume 1 0 and a union of two counties with one double the size and double the igidity of the othe (assumed to be = 2). o the Ricadian Wold in the whole J union in the R j = 0, j case, it is equied that > = 4 2 j j s j 1 20 inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1

20 Poposition 4: (i) Conside county j descibed by the Game of Chicken (11) and ensuing a Ricadian Wold. Afte joining a monetay union, if the degee of j s fee-iding is above a cetain (county specific) theshold, mj mj sj, v j, then j s accession leads to a deteioation of its stance fom P to A. (ii) The common cental bank s commitment theshold that ensues A is inceasing in the combined size of such counties with mj mj. If this size is sufficiently high then the Ricadian Wold is not achieved even if the bank is both fully patient and infinitely stongly committed, 1,. Poof: See Appendix D (on the website of this jounal). Intuitively, the theat of punishment by the common cental bank is no longe enough to discipline the govenment of a sufficiently small union membe county with a sufficiently high degee of fee-iding. It is no longe within the bank s powes to discouage such counties fom excesses though its actions. 21 This is because A becomes a stictly dominant stategy fo the fee-iding govenment in the undelying game and the scenaio fom its pespective changes fom the Game of Chicken to Neglect. In tems of claim (ii), the cental bank is woied about the cost of the policy conflict and theefoe once the combined active stance of policies eaches a cetain level, the bank will no longe play A as the esulting conflict would be too widespead and costly. The bank theefoe stats accommodating such policy, which leads to oveshooting of its optimal inflation level ove the long tem even if it is highly explicit. In such case, the cental bank s instument independence has been seiously compomised along the lines of the unpleasant monetaist aithmetic. 22 It emains to be seen (in the post ecovey situation) whethe such naative applies to the situation in the Euo aea. 7. Altenative Scenaios and the Real Wold 7.1 Altenative Scenaios It should by now be appaent that in the Symbiosis, Discipline, Tug-of-wa and Neglect scenaios dynamic commitment like static commitment will not alte the outcomes of the game. This is because the undelying nomal-fom game has a unique (and efficient) Nash equilibium with at least one playe having a stictly dominant stategy. In the Pue coodination scenaio, the condition fo selecting a Ricadian equilibium is the same, both qualitatively and quantitatively, as in the Game of Chicken. But even if it is not satisfied, the pobability of aiving in a non-ricadian egime is aguably lowe due to the focal point agument. 21 Geece pio to the global financial cisis comes to mind as an example of this type of behavio. 22 As Benanke (2005) agued: No monetay-policy egime, including inflation tageting, will succeed in educing inflation pemanently in the face of unsustainable fiscal policies lage and gowing deficits. inance a úvě-czech Jounal of Economics and inance, 64, 2014, no. 1 21

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